Combinatorics #4

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papgust: Community Manager; Posts: 1537; Joined: Mon Aug 10, 2009 6:10 pmFollowed by: 252 members; Thanked: 653 times

by papgust » Sun Oct 18, 2009 5:15 am

There are 5 blue socks, 4 red socks and 3 green socks in Debu's wardrobe. He has to select 4 socks from this set. In how many ways can he do so?

A. 245
B. 120
C. 495
D. 60

Solution and explanation pls?

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Source: Beat The GMAT — Problem Solving | Sun Oct 18, 2009 5:15 am

NikolayZ: Master | Next Rank: 500 Posts; Posts: 124; Joined: Thu Jun 18, 2009 5:33 am; Thanked: 35 times

by NikolayZ » Sun Oct 18, 2009 8:37 am

Hey again !
We have 12 total socks.
The guy must choose 4 whatever-color socks. (I'm curious why 4?!! =) )
it is combinatorics problem, since order does not matter.

So, the formula for this problem is C=(Total!)/(n!(total-n)!), where n is the number of socks Debu must select.

12!/4!*8!=495.
I hope i didn't mix smth up.

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papgust: Community Manager; Posts: 1537; Joined: Mon Aug 10, 2009 6:10 pmFollowed by: 252 members; Thanked: 653 times

by papgust » Sun Oct 18, 2009 10:58 pm

Thanks NikolayZ,

OA is 495. Thanks for the explanation

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gkumar: 2009 Beat The GMAT Scholarship Winner; Posts: 182; Joined: Sat Mar 22, 2008 7:35 pmFollowed by: 2 members; Thanked: 3 times; GMAT Score:700+

by gkumar » Mon Oct 19, 2009 2:40 am

If the socks of same color are considered different, then we can arrange in C(12,4) which comes out to be (C) 495 as said above.

But if the socks of the same color are equivalent and hence we can not tell one color sock from another sock of the same color, then the total number of ways is much less. I am not sure how to do this part, but another blog said 14 without explanation.

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JeffB: Master | Next Rank: 500 Posts; Posts: 153; Joined: Tue Jun 02, 2009 2:48 pm; Location: Ohio; Thanked: 16 times

by JeffB » Mon Oct 19, 2009 8:23 am

BTG Scholarship winner....Congratz.

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gkumar: 2009 Beat The GMAT Scholarship Winner; Posts: 182; Joined: Sat Mar 22, 2008 7:35 pmFollowed by: 2 members; Thanked: 3 times; GMAT Score:700+

by gkumar » Mon Oct 19, 2009 8:38 am

Regarding the number of DIFFERENT ways to get socks (e.g., one green sock is equivalent to another green sock), which is 14:
The number of ways three naturals number add up to 4 are: 4+0+0, 3+1+0, 2+2+0, 2+1+1. Since now order matters, we have to count how many different ways three number add up to 4 including order: 4+0+0, three ways; 3+1+0, six ways; 2+2+0, three ways; 2+1+1, three ways. So, in total, it would be 15; but we have counted even the GGGG configurations, which is not possible. so the answer is 14

(Thanks JeffB!)

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